It is often necessary to present calculations, or at least their outlines, in a proof. However, when I need to do it, if the calculation takes more than one line (or perhaps two at the most), I feel a bit uneasy, and have wrestled with trying to present it in a better way. On friday, I was browsing a journal, and came across a paper, in a subject that I was interested in, which contained a number of calculations which were at least a half a page of nothing but formulas. This made me shudder. Ultimately the reason why we should write papers is to foster insight, and I think that such presentations fail that test.

You are lucky to work in a subject where most calculations take only a couple of lines.
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Victor ProtsakMay 31 '10 at 7:14

Actually, calculations often do take up more than a couple of lines. However, that's the threshold at which I believe the calculations need to be broken up into conceptual pieces.
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Victor MillerMay 31 '10 at 19:24

Conceptual blocks, then. I wholeheartedly agree with the spirit of your comment, but some fields are luckier than others in this regard.
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Victor ProtsakJun 1 '10 at 5:40

6 Answers
6

The 1970 paper of Halmos entitled "how to write mathematics" is a bit old, preposterous to the TeX era, but I think that many of his advices still make sense today. To summarize what he says, "use well chosen words instead of plethora of symbols.".

"Think about the alphabet". And don't hesitate to associate a meaningful symbol to a piece of formula that makes sense by itself, so as to keep your calculation as compact as possible. In particular, when there is some constant at the end of the computation, that comes from an agregation of many constants appearing during the computation, just call it C through the whole calculation, and at the end, give its actual value. "The value of the constant C is..." (I think I read that trick in some paper by Krantz).

"Use words correctly". Explain what is going on. Irrelevancy should be avoided. Writing "Now applying the Cauchy-Schwarz inequality leads to...", and just giving the result, may be better than actually applying it in the middle of the computation without mentioning it.

Beware of too heavy use of formulae notations. Something like "Now applying (32), we get ..." is less helpful than "Let us apply the upper bound on the curvature that was obtained with the help of the Gauss-Bonnet theorem." Give meaningful names to important formulas in your paper instead of refering to them through numbers. The section entitled "resist symbols" in the paper of Halmos gives a few other tricks to replace heavy formalism by well worded sentences.

I think that in all writing, and especially in mathematical writing, signposting is extremely important. So a calculation-heavy proof should include a short summary, and then the calculations should be very clearly marked so that the reader knows that "here are the nitty gritties of the calculations, and here's where they stop, so you can skip them and pick up at the end".

For short papers, I like the following format. Begin with a very short introduction, setting the paper in context and giving a couple-sentence outline of the paper. In the second section, state all definitions and theorems precisely, but do not provide any proofs that are more than one or two sentences. Then the rest of the document, in as many sections as necessary, provides the detailed proofs, with all non-immediate calculations. The point is that most readers can read sections 1 and 2 and get everything out of the paper, and the only people who will read sections 3+ are: the reviewers (one hopes), and anyone trying to generalize the actual proof to a different setting.

Isn't the standard wisdom to omit those things that should be a routine check for a reader familiar enough with the topic? This reminds me of Serre's statement about a Bourbaki proof versus a proof. (The Bourbaki proof is for non-experts and the proof is for experts...) One should be writing to someone...and the level of detail should reflect who that someone is.